In seismic data acquisition, the seismic source is typically positioned at a selected shot location, and the seismic reflections of the shot are detected (the “shot record”) by receivers also located at selected locations. Then, the source and receivers are moved to different locations and the process repeated, and in this manner a seismic survey is taken of a selected subterranean region. Ideally for many seismic processing and interpretation objectives, the source and receiver locations would lie on a uniformly and densely sampled grid but this is difficult to achieve in common industry practice for many reasons including surface obstructions, currents, cable feathering, and acquisition cost. Survey economics mandate that the spacings be as large as will still yield the required detail in the survey results. The desired seismic reflections are wavefields that reflect once from a subterranean interface between regions with different acoustic properties (such as the upper surface of a petroleum reservoir), and then travel back to the surface to be detected by a receiver. These desired data are often obscured by undesired multiple reflections of seismic energy. Multiple suppression techniques exist for reducing this noise problem. Migration is a seismic data processing technique for moving the reflection events in recorded data to their true locations, which is necessary for non-horizontal subsurface reflectors.
Certain advanced seismic processing techniques such as three-dimensional surface-related multiple elimination (“3D SRME”) and shot-record wave-equation migration (“SRWEM”) work best if input data were regularly and densely sampled in the seismic survey that generated the data or as reconstructed by some interpolation technique. In particular, 3D SRME requires dense sampling of both shots and receivers, while SRWEM efficiency is a function of the number of shot records in the data set. Current marine acquisition geometries do not provide the data suitable or optimal for either one of the above processing algorithms. In the case of 3D SRME, shot (and to a lesser degree, streamer, i.e., receiver) spacing is too coarse for the algorithm to predict 3-D multiples correctly.
Currently, a number of interpolation techniques exist that attempt to interpolate missing data and to place irregularly sampled data on a fixed grid. Spatial aliasing, i.e., insufficient (less than two points per wavelength) sampling of data along a space axis, is a universal difficulty affecting all existing methods. Aliasing distortion can be seen in any motion picture where the wheels of a moving car are shown. One frame shows a wheel in a certain rotational position, but the next frame may show the wheel more than one full revolution later for example, or a little less than one revolution later, causing the non-physical effect so familiar to moviegoers in which wheel rotation is inconsistent with car motion. More frames per second would reduce this problem, and interpolation between existing frames is a possible way to achieve the same result. Although existing seismic data interpolation methods can interpolate data “beyond aliasing” under certain assumptions, their performance degrades as shot and receiver sampling become coarser and aliasing becomes more severe. Given that typical shot intervals in marine acquisition are hundreds of meters in the crossline (transverse to the boat movement) direction, there is currently no effective way of interpolating data to make it suitable for 3D SRME, which requires a data spacing of roughly 10-20 meters. Similarly, there is no known method capable of reconstructing the 360-degree receiver coverage that would be ideal for SRWEM.
Conceptually, if the subsurface properties (i.e., seismic wave velocities and densities) are known, one should be able to reconstruct seismic data at arbitrary locations through the well-known process of modeling. However, this is not easily achievable in practice since subsurface properties are typically not known with sufficient accuracy. A more practical approach would be to first migrate seismic data using an approximate velocity field and then de-migrate the image to emulate the desired surface acquisition. While migration-demigration would provide very accurate results even in the most complex geologic settings, the cost incurred would be prohibitive given computer resources currently available in the industry.
A more cost-effective approach is to map data to zero offset with the help of Dip Move-Out (“DMO”) and reconstruct the desired shot-receiver geometry by performing Inverse Common Offset DMO. (DMO processing is a widely known seismic processing operation to correct for the fact that, for a dipping reflection, the component traces of a common-midpoint gather do not involve a common reflecting point.) Although this approach makes certain simplifying assumptions about wavefield propagation, it has been used successfully to regularize marine data and serves as the basis of a technique known as Azimuth Move-Out (“AMO”); see Biondi, Fomel and Chemingui, “Azimuth moveout for 3-D prestack imaging,” Geophysics 63, 574-588 (1998). AMO is an efficient way of rotating irregular pre-stack data to a single common azimuth and offset. The key issue in the case of the shot record interpolation problem is that the output shot records comprise all possible azimuths and offsets, with nearly continuous azimuth angle variation between 0 and 360 degrees. If existing technology were applied, it would have been necessary to perform thousands of AMO runs to reconstruct all possible azimuths and offsets that may be present in a shot record, once again leading to a prohibitively expensive approach. Stolt provides an alternative in the form of a general data reconstruction method that can map arbitrary input shot geometry into a regular output shot geometry. (“Seismic data mapping and reconstruction,” Geophysics 67, 890-908 (2002).) The one-step shot continuation method proposed by Stolt can be very costly if applied to a conventional marine survey in order to reconstruct densely spaced shot records with dense receiver coverage, as required, for example, for 3D SRWEM.
A good overview of the existing interpolation methods can be found in “Comparisons of interpolation methods in the presence of aliased events” by R. Abma and N. Kabir, Expanded Abstracts, SEG (2003).